I am working on the proof of theorem 2 in Chapter 2 Evans. I understood everything except the last sensentence. "Finally note $\|u(\cdot,t)\|_{L^{\infty}}\le t\|f\|_{L^{\infty}}\to 0$."
Could anyone help elaborate how this could be deduced from the proof? (In particular, I have no clue how to expand things like $\|u(\cdot,t)\|_{L^{\infty}}$…)
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Calculating the norm $||u(x,t)||_{\infty}=||\int_0^t \int_{\mathbb R^n}\Phi(y,s)f(x-y, t-s)dyds||\leq\int_0^t|| \int_{\mathbb R^n}\Phi(y,s)f(x-y, t-s)dy||ds\leq\int_0^t \int_{\mathbb R^n}||\Phi(y,s)f(x-y, t-s)||dyds$. Then use that $|\int_{\mathbb R^n} \Phi|=1$ and you will find it.